Modern CT system generations Measurement of attenuation

Transcription

1 CT reconstruction repetition & hints Reconstruction in CT and hints to the assignments Jørgen Arendt Jensen October 4, 16 Center for Fast Ultrasound Imaging, Build 349 Department of Electrical Engineering Center for Fast Ultrasound Imaging Department of Electrical Engineering Filtered backprojection algorithm Radon transform Filtered backprojection algorithm Filters and their impulse responses Advise for the assignments Exercise 5 Reading material: Prince & Links Chapter 6 & 9 From: W. A. Kalender; Computed Tomography, Publicis, 5 Modern CT system generations Measurement of attenuation From: W. A. Kalender; Computed Tomography, Publicis, 5 From: W. A. Kalender; Computer Tomography, Publicis, 5 1

2 Parallel beam projection Parallel beam projection geometry From: W. A. Kalender; Computed Tomography, Publicis, 5 y y Point ψ φ θ x CT coordinate system x Patient coordinate system 6/x 7/x 8/x

3 9/x 1/x 11/x 1/x 3

4 Shepp-Logan phantom Note that the Shepp-Logan phantom is not in Hounsfield units, but the relative scaling is the same. 13/x 14/x Hounsfield units Note that the in-vivo data on the website is off-set by 1 HU (-1 is off-set to ) Fourier slice theorem From: W. A. Kalender; Computed Tomography, Publicis, 5 Demo in: for_13/matlab_demo/cd_demo 16/x 4

5 Filtered backprojection Influence from number of projections Perform for all projections: Make Fourier transform of projected data Apply filter in Fourier domain Make invers transform Back-project and sum with previous image 17/x 18/x Ram-Lak filter Transfer function: Transfer function of filters Ram-Lak " $ h(ω) = # %$ ρ, ρ B else Impulse response ( * * B k = * h(k) = ) B k odd * " π $ # k % * ' & * + * k even 19/x /x 5

6 Shepp_Logan filter Transfer function: " $ sin πρ $ ρ 4B, ρ B h(ω) = # πρ $ 4B $ % else Shepp-Logan filter Impulse response 8B h(k) = π (4k 1) 1/x /x Hanning weighted filter Hanning weighted filter 3/x 4/x 6

7 Filter transfer functions and impulse responses Comparison between filters 5/x 6/x Circular convolution Circular convolution Filter Filter.. h(n).1 h(n) Periodic time signal Periodic time signal g 1 (n) Resulting periodic time signal with overlap g 1 (n) Periodic time signal without overlap g 1 (n) 4 g (n) Sample number (n) 7/x 8/x 7

8 Circular convolution Shepp-Logan Relative y coordinate Ideal Shepp logan phantom, 51 x 51 pixels, Range: [ ] 3 1 Backprojection Projection Patient grid Relative x coordinate 9/x 3/x Reconstruction Exercise 5: Image processing Filtered backprojection algorithm and choices Next time: Relation between different reconstruction method and advise for assignments Advise for the assignments after the lectures Reading material: Prince & Links chapter 6-9, 1 & 13 3/x 8

9 1. Show Shepp-Logan phantom images.. Shepp-Logan phantom gray scale mapping. 3. Clinical images of the brain and its gray scale mapping. 4. Make a two-dimensional Fourier transform of the sh_black image, and make a mesh plot of the amplitude spectrum with the command mesh. Plot the spectrum with the correct spatial frequency axis. Study the symmetry relations for the Fourier transform. 5. Make a low-pass filter with a circularly symmetric transfer function that removes all frequencies above a value of 116 m Use an edge enhancement filter given as [ ; ; ] to enhance the edges in the image sh_black. 7. Try the above mentioned image processing on the clinical images downloaded previously. 33/x 9